This gives the approximate coverage of the syllabus, lecture by lecture. Topics/lectures in green are not yet covered.
Introductory lectures (Chapter 1)
1. What are bifurcations? Examples. Fold.
2. Zeeman Catastrophe Machine. Evolutes
3. Evolute (ctd), Pitchfork.
PART I: Catastrophe Theory
Chapter 2
4. Families of functions, degenerate critical points.
5. Cusp catastrophe.
Chapter 3: Some algebra
6, 7. Ring of germs of smooth functions; its maximal ideal, Hadamard's lemma.
8. Newton Diagram, finite codimension ideals, cobasis.
9a. Nakayama's Lemma ...
Chapter 4: Right Equivalence
9b. Diffeomorphisms, changes of coordinates and inverse function theorem.
10. Right equivalence,
11. Jacobian ideals/modules. Codimension of critical points.
12. Morse Lemma. Corank of a critical point and the Splitting Lemma
-- Reading Week --
13.
xx. Proof of inverse function theorem by homotopy method. xx. Immersions and submersions
xx. Submanifolds and parametrizations.
xx. Linearly adapted coordinates and Lyapounov-Schmidt reduction; germs.
Chapter 5: Finite determinacy
14. \(\mathcal{R}\)-trivial families and \(\mathcal{R}\)-tangent space; vector fields along \(f\)
15. Finite determinacy theorem + Examples.
16. Refinements of Theorem
17. Proof of theorem.
Chapter 6: Classification
17. Classification of critical points of functions
18. Classification ctd -- corank 1 critical points and classification of binary cubics.
19. Classification for corank 2 critical points
Chapter 7: Unfoldings of function germs
20. Families, equivalence,
21. Unfoldings
22. Versality theorem for \(\mathcal{R}\)-equivalence ...
PART II: Singularity Theory
Chapter 8: Singularities of Maps
23. Bifurcation problems - what are they?
24. Contact equivalence. Tangent spaces.
25. Classification of corank 1 singularities (the \(A_k\) family) and corank 2.
Chapter 9: Unfoldings & versality
26. Families of maps as unfoldings. Equivalent and versal unfoldings.
27. Versality theorem (no proof) and examples.